Geodesic
Galois groups of multivariate Tutte polynomials
Journal of Algebraic Combinatorics, Tome 36 (2012) no. 2, pp. 223-230.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The multivariate Tutte polynomial $\hat{Z}_{M}$ of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable v $_{ e }$ to each element e of the ground set E. It encodes the full structure of M. Let v=v $_{ e }}_{ e\in E }$, let K be an arbitrary field, and suppose M is connected. We show that $\hat{Z}_{M}$ is irreducible over $K(v)$, and give three self-contained proofs that the Galois group of $\hat{Z}_{M}$ over $K(v)$ is the symmetric group of degree n, where n is the rank of M. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.
Keywords: tutte polynomial, multivariate tutte polynomial, matroids, graphs, Galois theory 
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     author = {Bohn, Adam and Cameron, Peter J. and M\"uller, Peter},
     title = {Galois groups of multivariate {Tutte} polynomials},
     journal = {Journal of Algebraic Combinatorics},
     pages = {223--230},
     publisher = {mathdoc},
     volume = {36},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2012__36_2_a5/}
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Bohn, Adam; Cameron, Peter J.; Müller, Peter. Galois groups of multivariate Tutte polynomials. Journal of Algebraic Combinatorics, Tome 36 (2012) no. 2, pp. 223-230. http://geodesic.mathdoc.fr/item/JAC_2012__36_2_a5/